## Towards the Notion of Stability of Approximation for Hard Optimization Tasks and the Traveling Salesman Problem (1999)

Venue: | Comput. Sci |

Citations: | 16 - 2 self |

### BibTeX

@ARTICLE{Böckenhauer99towardsthe,

author = {Hans-Joachim Böckenhauer and Juraj Hromkovic and Ralf Klasing and Sebastian Seibert and Walter Unger},

title = {Towards the Notion of Stability of Approximation for Hard Optimization Tasks and the Traveling Salesman Problem},

journal = {Comput. Sci},

year = {1999},

volume = {285},

pages = {3--24}

}

### OpenURL

### Abstract

The investigation of the possibility to efficiently compute approximations of hard optimization problems is one of the central and most fruitful areas of current algorithm and complexity theory. The aim of this paper is twofold. First, we introduce the notion of stability of approximation algorithms. This notion is shown to be of practical as well as of theoretical importance, especially for the real understanding of the applicability of approximation algorithms and for the determination of the border between easy instances and hard instances of optimization problems that do not admit polynomial-time approximation. Secondly, we apply our concept to the study of the traveling salesman problem. We show how to modify the Christofides algorithm for \Delta-TSP to obtain efficient approximation algorithms with constant approximation ratio for every instance of TSP that violates the triangle inequality by a multiplicative constant factor. This improves the result of Andreae and Ba...

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Citation Context ...problem does not admit an efficiently computable optimal solution, is there a possibility to efficiently compute at least an approximation of the optimal solution? Several researchers [Jo74], [Lo75], =-=[Chr76]-=-, [IK75] provided already in the middle of the seventies a positive answer for some optimization problems. It is a fascinating effect if one can jump from exponential complexity (a huge inevitable amo... |

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Citation Context ...ation algorithms with constant approximation ratio for every instance of TSP that violates the triangle inequality by a multiplicative constant factor. This improves the result of Andreae and Bandelt =-=[AB95]-=-. Keywords: Stability of approximation, Traveling Salesman Problem 1 Introduction Immediately after introducing NP-hardness (completeness) [Co71] as a concept for proving intractability of computing p... |

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Citation Context ...riginal 2-approximation algorithm for \Delta-TSP. Note that, after this paper was written, we got the information about the independent, unpublished result of Bender and Chekuri, accepted for WADS'99 =-=[BC99]-=-. They designed a 4fi-approximation algorithm which can be seen as a modification of the 2-approximation algorithm for \Delta fi -TSP. Despite this nice result, there are three reasons to consider our... |

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Citation Context ...actor. This improves the result of Andreae and Bandelt [AB95]. Keywords: Stability of approximation, Traveling Salesman Problem 1 Introduction Immediately after introducing NP-hardness (completeness) =-=[Co71]-=- as a concept for proving intractability of computing problems, the following question has been posed: If an optimization problem does not admit an efficiently computable optimal solution, is there a ... |

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Citation Context ...ven in the last section. In the next section we show a transparent application of our concept for TSP. 3 Stability of Approximation Algorithms and TSP We consider the well-known TSP problem (see e.g. =-=[LLRS85]-=-) that is in its general form very hard for approximation. But if one considers complete graphs in which the triangle inequality holds, then we have a 3 2 -approximation algorithm due to Christofides ... |

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Citation Context ...ization problem does not admit an efficiently computable optimal solution, is there a possibility to efficiently compute at least an approximation of the optimal solution? Several researchers [Jo74], =-=[Lo75]-=-, [Chr76], [IK75] provided already in the middle of the seventies a positive answer for some optimization problems. It is a fascinating effect if one can jump from exponential complexity (a huge inevi... |